PlanetMath: Bernoulli number

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Bernoulli number

(Definition)

Let be the th Bernoulli polynomial. Then the th Bernoulli number is

$\displaystyle B_r := B_r(0).$

This means, in particular, that the Bernoulli numbers are given by an exponential generating function in the following way:

$\displaystyle \sum_{r=0}^{\infty} B_r \frac{y^r}{r!} = \frac{y}{e^y-1}$

and, in fact, the Bernoulli numbers are usually defined as the coefficients that appear in such expansion.

Observe that this generating function can be rewritten:

$\displaystyle \frac{y}{e^y-1} = \frac{y}{2}\frac{e^y+1}{e^y-1} - \frac{y}{2} = (y/2)(\operatorname{tanh}(y/2) -1).$

Since $\operatorname{tanh}$ is an odd function, one can see that $B_{2r+1}=0$ for $r \geq 1$ . Numerically, $B_0 = 1, B_1 = -\frac{1}{2}, B_2 = \frac{1}{6}, B_4 = -\frac{1}{30}, \cdots$

These combinatorial numbers occur in a number of contexts; the most elementary is perhaps that they occur in the formulas for the sum of the th powers of the first positive integers. They also occur in the Maclaurin expansion for the tangent function and in the Euler-Maclaurin summation formula.

"Bernoulli number" is owned by alozano. [ full author list (3) | owner history (2) ]

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th powers of the first $n$

positive integers, Euler-Maclaurin summation formula, values of the Riemann zeta function in terms of Bernoulli numbers, Taylor series via division

Keywords:

number theory

Attachments:: congruence of Clausen and von Staudt (Theorem) by alozano
Kummer's congruence (Theorem) by alozano
the odd Bernoulli numbers are zero (Theorem) by alozano
recurrence formula for Bernoulli numbers (Derivation) by rm50

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Cross-references: Euler-Maclaurin summation formula, function, tangent, occur in, numbers, odd function, coefficients, generating function, exponential, Bernoulli polynomial
There are 24 references to this entry.

This is version 9 of Bernoulli number, born on 2001-10-15, modified 2007-12-20.
Object id is 219, canonical name is BernoulliNumber.
Accessed 7634 times total.

Classification:

AMS MSC:

11B68 (Number theory :: Sequences and sets :: Bernoulli and Euler numbers and polynomials)

Pending Errata and Addenda

None.[ View all 3 ]

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